Flip Signatures

TL;DR

This paper introduces flip signature as a new invariant for $D_{ fty}$-topological Markov chains, demonstrating its effectiveness in distinguishing non-conjugate systems with natural $D_{ fty}$-actions.

Contribution

The paper defines flip signature, proves it is a $D_{ fty}$-conjugacy invariant, and applies it to differentiate specific $D_{ fty}$-actions on symbolic systems.

Findings

Flip signature is a $D_{ fty}$-conjugacy invariant.

Ashley’s eight-by-eight and full two-shift are not $D_{ fty}$-conjugate.

Introduces $D_{ fty}$-shift equivalence and discusses Lind zeta function.

Abstract

A $D_{\infty}$-topological Markov chain is a topological Markov chain provided with an action of the infinite dihedral group $D_{\infty}$. It is defined by two zero-one square matrices $A$ and $J$ satisfying $AJ=JA^{\textsf{T}}$ and $J^2=I$. Flip signature is obtained from symmetric bilinear forms with respect to $J$ on the eventual kernel of $A$. We modify Williams' decomposition theorem to prove flip signature is a $D_{\infty}$-conjugacy invariant. We introduce natural $D_{\infty}$-actions on Ashley's eight-by-eight and the full two-shift. The Flip signatures show that Ashley's eight-by-eight and the full two-shift equipped with the natural $D_{\infty}$-actions are not $D_{\infty}$-conjugate. We also discuss the notion of $D_{\infty}$-shift equivalence and the Lind zeta function.